Matrix multiplication and row operations

Matrix multiplication and row operations

Compare these two multiplications:

\begin{align*} \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} e \\ f \end{pmatrix}\\ \begin{pmatrix} c & d \\ a & b \\ \end{pmatrix} \begin{pmatrix} e \\ f \end{pmatrix}\\ \end{align*}

You can see the row changes in the first matrix will result in the same row changes in the end result, the reason is:

\begin{align*} (EA)B = E(AB) \end{align*}

Where E is an elementary matrix.